Tameness conjecture for generic codimension-one degeneracies

Let $\v \in \operatorname{Vect}^{*}_k(\Omega)$, where $\Omega$ is an open subset of $S^2$ or $S^2$ itself, and suppose that $v$ contains only a generic degeneracy of codimension $1$. Tameness conjecture. The vector field $v$ is tame, meaning that nearby structurally unstable vector fields orbitally topologically equivalent to $v$ form a finite-codimensional Banach submanifold. This conjecture would provide the finite-dimensional manifold structure needed to study generic unfoldings and their SET property.